Beyond identical utilities: buyer utility functions and fair allocations

TL;DR

This paper introduces a polynomial-time algorithm for fair and efficient allocations under buyer utility functions, achieving EF1, PO, and EFX properties, advancing computational methods in fair division.

Contribution

It presents the first polynomial-time algorithm that maximizes social welfare while ensuring EF1, PO, and EFX for buyer utility functions, a special case of additive utilities.

Findings

Algorithm achieves EF1 and PO in polynomial time for buyer utility functions.

Modification yields allocations that are envy-free up to any positively valued good (EFX).

Provides practical methods for fair division with computational efficiency.

Abstract

The problem of finding envy-free allocations of indivisible goods can not always be solved; therefore, it is common to study some relaxations such as envy-free up to one good (EF1). Another property of interest for efficiency of an allocation is the Pareto Optimality (PO). Under additive utility functions, it is possible to find allocations EF1 and PO using Nash social welfare. However, to find an allocation that maximizes the Nash social welfare is a computationally hard problem. In this work we propose a polynomial time algorithm which maximizes the utilitarian social welfare and at the same time produces an allocation which is EF1 and PO in a special case of additive utility functions called buyer utility functions. Moreover, a slight modification of our algorithm produces an allocation which is envy-free up to any positively valued good (EFX).